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Scale Invariance without Inflation?

C. Armendariz-Picon, Eugene A. Lim

TL;DR

This work proposes a non-inflationary mechanism to seed a scale-invariant primordial perturbation spectrum by allowing the sound speed to decay in an expanding isentropic fluid. Using the Mukhanov variable framework, the authors derive conditions under which perturbations can freeze behind the sound horizon and obtain a scale-invariant tilt for a polytropic index $\alpha$, with $n_s-1 = (\alpha-6)/(\alpha-1)$ and scale invariance at $\alpha=6$. The concrete toy model demonstrates this mechanism but reveals a tension: matching the observed amplitude across about three decades in $k$-space requires a seeding duration that conflicts with perturbativity and data unless a subsequent inflationary phase or another boosting process amplifies the seeded spectrum. They also provide a microphysical realization via a k-essence Lagrangian $p(X)$ that reproduces the required equation of state. Overall, while offering a novel seeding mechanism, the approach faces substantial challenges to reproduce cosmological perturbations without inflation.

Abstract

We propose a new alternative mechanism to seed a scale invariant spectrum of primordial density perturbations that does not rely on inflation. In our scenario, a perfect fluid dominates the early stages of an expanding, non-inflating universe. Because the speed of sound of the fluid decays, perturbations are left frozen behind the sound horizon, with a spectral index that depends on the fluid equation of state. We explore here a toy model that realizes this idea. Although the model can explain an adiabatic, Gaussian, scale invariant spectrum of primordial perturbations, it turns out that in its simplest form it cannot account for the observed amplitude of the primordial density perturbations.

Scale Invariance without Inflation?

TL;DR

This work proposes a non-inflationary mechanism to seed a scale-invariant primordial perturbation spectrum by allowing the sound speed to decay in an expanding isentropic fluid. Using the Mukhanov variable framework, the authors derive conditions under which perturbations can freeze behind the sound horizon and obtain a scale-invariant tilt for a polytropic index , with and scale invariance at . The concrete toy model demonstrates this mechanism but reveals a tension: matching the observed amplitude across about three decades in -space requires a seeding duration that conflicts with perturbativity and data unless a subsequent inflationary phase or another boosting process amplifies the seeded spectrum. They also provide a microphysical realization via a k-essence Lagrangian that reproduces the required equation of state. Overall, while offering a novel seeding mechanism, the approach faces substantial challenges to reproduce cosmological perturbations without inflation.

Abstract

We propose a new alternative mechanism to seed a scale invariant spectrum of primordial density perturbations that does not rely on inflation. In our scenario, a perfect fluid dominates the early stages of an expanding, non-inflating universe. Because the speed of sound of the fluid decays, perturbations are left frozen behind the sound horizon, with a spectral index that depends on the fluid equation of state. We explore here a toy model that realizes this idea. Although the model can explain an adiabatic, Gaussian, scale invariant spectrum of primordial perturbations, it turns out that in its simplest form it cannot account for the observed amplitude of the primordial density perturbations.

Paper Structure

This paper contains 7 sections, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formalism
  3. Decaying sound speed
  4. A simple model
  5. Observational constraints
  6. A Microscopic Description
  7. Conclusions

Figures (1)

  • Figure 1: A schematic plot of the evolution of the different length scales in our scenario. During seeding, the physical length $\lambda$ (thick line) crosses the sound horizon $c_s/H$ (continuous line). Later, during inflation, $\lambda$ exits the Hubble radius $1/H$ (dot-dashed line).